This document discusses integration, which is the inverse operation of differentiation. It begins by explaining that integration finds the original function given its derivative, with the addition of a constant of integration. It then provides examples of basic integration techniques using a table of integrals. The document also outlines some rules for integrating sums and constant multiples of functions. Finally, it gives an example of using integration to solve an engineering problem involving the electric potential of a charged sphere.

1. ContentsContents 1313
Integration
13.1 Basic Concepts of Integration 2
13.2 Definite Integrals 14
13.3 The Area Bounded by a Curve 24
13.4 Integration by Parts 33
13.5 Integration by Substitution and Using Partial Fractions 40
13.6 Integration of Trigonometric Functions 48
Learning
In this Workbook you will learn about integration and about some of the common techniques
employed to obtain integrals. You will learn that integration is the inverse operation to
differentiation and will also appreciate the distinction between a definite and an indefinite
integral. You will understand how a definite integral is related to the area under a curve.
You will understand how to use the technique of integration by parts to obtain integrals
involving the product of functions. You will also learn how to use partial fractions and
trigonometric identities in integration.
outcomes

2. Basic Concepts
of Integration
13.1
Introduction
When a function f(x) is known we can differentiate it to obtain its derivative
df
dx
. The reverse process
is to obtain the function f(x) from knowledge of its derivative. This process is called integration.
Applications of integration are numerous and some of these will be explored in subsequent Sections.
First, what is important is to practise basic techniques and learn a variety of methods for integrating
functions.
Prerequisites
Before starting this Section you should . . .
• thoroughly understand the various techniques
of differentiation
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$
%
Learning Outcomes
On completion you should be able to . . .
• evaluate simple integrals by reversing the
process of differentiation
• use a table of integrals
• explain the need for a constant of integration
when finding indefinite integrals
• use the rules for finding integrals of sums of
functions and constant multiples of functions
2 HELM (2008):
Workbook 13: Integration

3. ®
1. Integration as differentiation in reverse
Suppose we differentiate the function y = x2
. We obtain
dy
dx
= 2x. Integration reverses this process
and we say that the integral of 2x is x2
. Pictorially we can regard this as shown in Figure 1:
x2
2x
differentiate
integrate
Figure 1
The situation is just a little more complicated because there are lots of functions we can differentiate
to give 2x. Here are some of them: x2
+ 4, x2
− 15, x2
+ 0.5
All these functions have the same derivative, 2x, because when we differentiate the constant term we
obtain zero. Consequently, when we reverse the process, we have no idea what the original constant
term might have been. So we include in our answer an unknown constant, c say, called the constant
of integration. We state that the integral of 2x is x2
+ c.
When we want to differentiate a function, y(x), we use the notation
d
dx
as an instruction to differ-
entiate, and write
d
dx
y(x) . In a similar way, when we want to integrate a function we use a special
notation: y(x) dx.
The symbol for integration, , is known as an integral sign. To integrate 2x we write
2x dx = x2 + c
integral
sign
this term is
called the
integrand there must always be a
term of the form dx
constant of integration
Note that along with the integral sign there is a term of the form dx, which must always be written,
and which indicates the variable involved, in this case x. We say that 2x is being integrated with
respect to xxx. The function being integrated is called the integrand. Technically, integrals of this
sort are called indefinite integrals, to distinguish them from definite integrals which are dealt with
subsequently. When you find an indefinite integral your answer should always contain a constant of
integration.
Exercises
1. (a) Write down the derivatives of each of: x3
, x3
+ 17, x3
− 21
(b) Deduce that 3x2
dx = x3
+ c.
2. Explain why, when finding an indefinite integral, a constant of integration is always needed.
HELM (2008):
Section 13.1: Basic Concepts of Integration
3

4. Answers
1. (a) 3x2
, 3x2
, 3x2
(b) Whatever the constant, it is zero when differentiated.
2. Any constant will disappear (i.e. become zero) when differentiated so one must be reintroduced
to reverse the
process.
2. A table of integrals
We could use a table of derivatives to find integrals, but the more common ones are usually found
in a ‘Table of Integrals’ such as that shown below. You could check the entries in this table using
your knowledge of differentiation. Try this for yourself.
Table 1: Integrals of Common Functions
function indefinite integral
f(x) f(x) dx
constant, k kx + c
x 1
2
x2
+ c
x2 1
3
x3
+ c
xn xn+1
n + 1
+ c, n = −1
x−1
(or
1
x
) ln |x| + c
cos x sin x + c
sin x − cos x + c
cos kx
1
k
sin kx + c
sin kx −
1
k
cos kx + c
tan kx
1
k
ln | sec kx|+c
ex
ex
+ c
e−x
−e−x
+ c
ekx 1
k
ekx
+ c
When dealing with the trigonometric functions the variable x must always be measured in radians
and not degrees. Note that the fourth entry in the Table, for xn
, is valid for any value of n, positive
or negative, whole number or fractional, except n = −1. When n = −1 use the fifth entry in the
Table.
4 HELM (2008):
Workbook 13: Integration

5. ®
Example 1
Use Table 1 to find the indefinite integral of x7
: that is, find x7
dx
Solution
From Table 1 note that xn
dx =
xn+1
n + 1
+ c. In words, this states that to integrate a power
of x, increase the power by 1, and then divide the result by the new power. With n = 7 we find
x7
dx =
1
8
x8
+ c
Example 2
Find the indefinite integral of cos 5x: that is, find cos 5x dx
Solution
From Table 1 note that cos kx dx =
sin kx
k
+ c
With k = 5 we find cos 5x dx =
1
5
sin 5x + c
In Table 1 the independent variable is always given as x. However, with a little imagination you will
be able to use it when other independent variables are involved.
Example 3
Find cos 5t dt
Solution
We integrated cos 5x in the previous example. Now the independent variable is t, so simply use
Table 1 and replace every x with a t. With k = 5 we find
cos 5t dt =
1
5
sin 5t + c
It follows immediately that, for example,
cos 5ω dω =
1
5
sin 5ω + c, cos 5u du =
1
5
sin 5u + c and so on.
HELM (2008):
Section 13.1: Basic Concepts of Integration
5

6. Example 4
Find the indefinite integral of
1
x
: that is, find
1
x
dx
Solution
This integral deserves special mention. You may be tempted to try to write the integrand as x−1
and use the fourth row of Table 1. However, the formula xn
dx =
xn+1
n + 1
+ c is not valid when
n = −1 as Table 1 makes clear. This is because we can never divide by zero. Look to the fifth
entry of Table 1 and you will see x−1
dx = ln |x| + c.
Example 5
Find 12 dx and 12 dt
Solution
In this Example we are integrating a constant, 12. Using Table 1 we find
12 dx = 12x + c Similarly 12 dt = 12t + c.
Task
Find t4
dt
Your solution
Answer
t4
dt =
1
5
t5
+ c.
6 HELM (2008):
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7. ®
Task
Find
1
x5
dx using the laws of indices to write the integrand as x−5
and then use
Table 1:
Your solution
Answer
−
1
4
x−4
+ c = −
1
4x4
+ c.
Task
Find e−2x
dx using the entry in Table 1 for integrating ekx
:
Your solution
Answer
With k = −2, we have e−2x
dx = −
1
2
e−2x
+ c.
Exercises
1. Integrate each of the following functions with respect to x:
(a) x9
, (b) x1/2
, (c) x−3
, (d) 1/x4
, (e) 4, (f)
√
x, (g) e4x
2. Find (a) t2
dt, (b) 6 dt, (c) sin 3t dt, (d) e7t
dt.
Answers
1 (a)
1
10
x10
+ c, (b)
2
3
x3/2
+ c, (c) −
1
2
x−2
+ c, (d) −
1
3
x−3
+ c, (e) 4x + c,
(f) same as (b), (g)
1
4
e4x
+ c
2. (a)
1
3
t3
+ c, (b) 6t + c, (c) −
1
3
cos 3t + c, (d)
1
7
e7t
+ c
HELM (2008):
Section 13.1: Basic Concepts of Integration
7

8. 3. Some rules of integration
To enable us to find integrals of a wider range of functions than those normally given in a table of
integrals we can make use of the following rules.
The integral of kkk fff(xxx) where kkk is a constant
A constant factor in an integral can be moved outside the integral sign as follows:
Key Point 1
k f(x) dx = k f(x) dx
Example 6
Find the indefinite integral of 11x2
: that is, find 11x2
dx
Solution
11x2
dx = 11 x2
dx = 11
x3
3
+ c =
11x3
3
+ K where K is a constant.
Example 7
Find the indefinite integral of −5 cos x; that is, find −5 cos x dx
Solution
−5 cos x dx = −5 cos x dx = −5 (sin x + c) = −5 sin x + K where K is a constant.
8 HELM (2008):
Workbook 13: Integration

9. ®
The integral of fff(xxx) +++ ggg(xxx) and of fff(xxx) −−− ggg(xxx)
When we wish to integrate the sum or difference of two functions, we integrate each term separately
as follows:
Key Point 2
[ f(x) + g(x) ] dx = f(x) dx + g(x) dx
[ f(x) − g(x) ] dx = f(x) dx − g(x) dx
Example 8
Find (x3
+ sin x) dx
Solution
(x3
+ sin x) dx = x3
dx + sin x dx =
1
4
x4
− cos x + c
Note that only a single constant of integration is needed.
Task
Find (3t4
+
√
t) dt
Use Key Points 1 and 2:
Your solution
Answer
3
5
t5
+
2
3
t3/2
+ c
HELM (2008):
Section 13.1: Basic Concepts of Integration
9

10. Task
The hyperbolic sine and cosine functions, sinh x and cosh x, are defined as follows:
sinh x =
ex
− e−x
2
cosh x =
ex
+ e−x
2
Note that they are combinations of the exponential functions ex
and e−x
.
Find the indefinite integrals of sinh x and cosh x.
Your solution
sinh x dx =
ex
− e−x
2
dx =
cosh x dx =
ex
+ e−x
2
dx =
Answer
sinh x dx =
1
2
ex
dx −
1
2
e−x
dx =
1
2
ex
+
1
2
e−x
+ c =
1
2
ex
+ e−x
+ c = cosh x + c.
Similarly cosh x dx = sinh x + c.
Further rules for finding more complicated integrals are dealt with in subsequent Sections.
10 HELM (2008):
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11. ®
Engineering Example 1
Electrostatic charge
Introduction
Electrostatic charge is important both where it is wanted, as in the electrostatic precipitator plate
systems used for cleaning gases, and where it is unwanted, such as when charge builds up on moving
belts. This Example is concerned with a charged object with a particular idealised shape - a sphere.
However, similar analytical calculations can be carried out for certain other shapes and numerical
methods can be used for more complicated shapes.
The electric field at all points inside and outside a charged sphere is given by
E(r) =
Qr
4πε0a3
if r a (1a)
E(r) =
Q
4πε0r2
if r ≥ a (1b)
where ε0 is the permittivity of free space, Q is the total charge, a is the radius of the sphere, and r
is the radial distance between the centre of the sphere and a point of observation (see Figure 2).
Charged sphere
Spherical surface S
a
O
r
Figure 2: Geometry and symbols associated with the charged sphere
The electric field associated with electrostatic charge has a scalar potential. The electric field defined
by (1a) and (1b) shows only a radial dependence of position. Therefore, the electric scalar potential
V (r) is related to the field E(r) by
E(r) = −
dV
dr
. (2)
Problem in words
A sphere is charged with a uniform density of charge and no other charge is present outside the
sphere in space. Determine the variation of electric potential with distance from the centre of the
sphere.
Mathematical statement of problem
Determine the electric scalar potential as a function of r, V (r), by integrating (2).
Mathematical analysis
Equation (2) yields V (r) as the negative of the indefinite integral of E(r).
HELM (2008):
Section 13.1: Basic Concepts of Integration
11

12. − dV = E(r) dr. (3)
Using (1a) and (1b) with (3) leads to
V (r) = −
Q
4πε0a3
r dr if r a (4a)
V (r) = −
Q
4πε0
dr
r2
if r ≥ a (4b)
Using the facts that r dr = r2
/2 + c1 and
dr
r2
= −
1
r
+ c2,
(4a) and (4b) become
V (r) = −
Qr2
8πε0a3
+ c1 if r a (5a)
V (r) =
Q
4πε0r
+ c2 if r ≥ a (5b)
The integration constant c2 can be determined by assuming that the electric potential is zero at an
infinite distance from the sphere:
lim
r→∞
[V (r)] = 0 ⇒ lim
r→∞
−
Q
4πε0r
+ c2 = 0 ⇒ c2 = 0.
The constant c1 can be determined by assuming that the potential is continuous at r = a.
From equation (5a)
V (a) = −
Qa2
8πε0a3
+ c1
From equation (5b)
V (a) =
Q
4πε0a
Hence
c1 =
Q
4πε0a
+
2Q
8πε0a
=
3Q
8πε0a
.
Substituting for c1 in (5), the electric potential is obtained for all space is:
V (r) =
Q
4πε0
3a2
− r2
2a3
if r a.
V (r) =
Q
4πε0r
if r ≥ a
Interpretation
The potential of the electrostatic field outside a charged sphere varies inversely with distance from
the centre of the sphere. Inside the sphere, the electrostatic potential varies with the square of the
distance from the centre.
An Engineering Exercise in 29.3 derives the corresponding expressions for the variation of the
electrostatic field and an Engineering Exercise in 27.4 calculates the potential energy due to
the charged sphere.
12 HELM (2008):
Workbook 13: Integration

13. ®
Exercises
1. Find (2x − ex
) dx
2. Find 3e2x
dx
3. Find
1
3
(x + cos 2x) dx
4. Find 7x−2
dx
5. Find (x + 3)2
dx, (be careful!)
Answers
1. x2
− ex
+ c
2.
3
2
e2x
+ c
3.
1
6
x2
+
1
6
sin 2x + c
4. −
7
x
+ c
5.
1
3
x3
+ 3x2
+ 9x + c
HELM (2008):
Section 13.1: Basic Concepts of Integration
13